85. Find the volume of the parallelepiped with edges given by position vectors ⟨5,0,0⟩{\displaystyle \langle 5,0,0\rangle }, ⟨1,4,0⟩{\displaystyle \langle 1,4,0\rangle }, and ⟨2,2,7⟩{\displaystyle \langle 2,2,7\rangle }

140{\displaystyle 140}

86. A wrench has a pivot at the origin and extends along the x-axis. Find the magnitude and the direction of the torque at the pivot when the force F=⟨1,2,3⟩{\displaystyle \mathbf {F} =\langle 1,2,3\rangle } is applied to the wrench n units away from the origin.

Once expressed in component form, both sides evaluate to u1v2w3−u1v3w2+u2v3w1−u2v1w3+u3v1w2−u3v2w1{\displaystyle u_{1}v_{2}w_{3}-u_{1}v_{3}w_{2}+u_{2}v_{3}w_{1}-u_{2}v_{1}w_{3}+u_{3}v_{1}w_{2}-u_{3}v_{2}w_{1}}

225. The volume of a pyramid with a square base is V=13x2h{\displaystyle V={\frac {1}{3}}x^{2}h}, where x is the side of the square base and h is the height of the pyramid. Suppose that x(t)=tt+1{\displaystyle \displaystyle x(t)={\frac {t}{t+1}}} and h(t)=1t+1{\displaystyle \displaystyle h(t)={\frac {1}{t+1}}} for t≥0.{\displaystyle t\geq 0.} Find V′(t).{\displaystyle V'(t).}

424. Find the work required to move an object from (1,1,1) to (8,4,2) along a straight line in the force field F=⟨x,y,z⟩x2+y2+z2{\displaystyle \displaystyle \mathbf {F} ={\frac {\langle x,y,z\rangle }{x^{2}+y^{2}+z^{2}}}}

460. Evaluate the circulation of the field F=⟨2xy,x2−y2⟩{\displaystyle \mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle } over the boundary of the region above y=0 and below y=x(2-x) in two different ways, and compare the answers.

0{\displaystyle 0}

461. Evaluate the circulation of the field F=⟨0,x2+y2⟩{\displaystyle \mathbf {F} =\langle 0,x^{2}+y^{2}\rangle } over the unit circle centered at the origin in two different ways, and compare the answers.

0{\displaystyle 0}

462. Evaluate the flux of the field F=⟨y,−x⟩{\displaystyle \mathbf {F} =\langle y,-x\rangle } over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.

520. Use a surface integral to evaluate the circulation of the field F=⟨x2−z2,y,2xz⟩{\displaystyle \mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle } on the boundary of the plane z=4−x−y{\displaystyle z=4-x-y} in the first octant.

−1283{\displaystyle {\frac {-128}{3}}}

521. Use a surface integral to evaluate the circulation of the field F=⟨y2,−z2,x⟩{\displaystyle \mathbf {F} =\langle y^{2},-z^{2},x\rangle } on the circle r(t)=⟨3cos⁡(t),4cos⁡(t),5sin⁡(t)⟩.{\displaystyle \mathbf {r} (t)=\langle 3\cos(t),4\cos(t),5\sin(t)\rangle .}

523. Use a line integral to find ∬S(∇×F)⋅ndS{\displaystyle \iint _{S}(\nabla \times F)\cdot \mathbf {n} dS}
where F=⟨2y,−z,x−y−z⟩{\displaystyle \mathbf {F} =\langle 2y,-z,x-y-z\rangle }, S{\displaystyle S} is the part of the sphere x2+y2+z2=25{\displaystyle x^{2}+y^{2}+z^{2}=25} for 3≤z≤5{\displaystyle 3\leq z\leq 5}, and n{\displaystyle \mathbf {n} } points in the direction of the z-axis.

543. F=⟨x,y,z⟩{\displaystyle \mathbf {F} =\langle x,y,z\rangle }, S{\displaystyle S} is the surface of the region bounded by the paraboloid z=4−x2−y2{\displaystyle z=4-x^{2}-y^{2}} and the xy-plane.

24π{\displaystyle 24\pi }

544. F=⟨z−x,x−y,2y−z⟩{\displaystyle \mathbf {F} =\langle z-x,x-y,2y-z\rangle }, S{\displaystyle S} is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.

−224π{\displaystyle -224\pi }

545. F=⟨x,2y,3z⟩{\displaystyle \mathbf {F} =\langle x,2y,3z\rangle }, S{\displaystyle S} is the boundary of the region between the cylinders x2+y2=1{\displaystyle x^{2}+y^{2}=1} and x2+y2=4{\displaystyle x^{2}+y^{2}=4} and cut off by planes z=0{\displaystyle z=0} and z=8{\displaystyle z=8}